The invention relates generally to control systems, and more particularly to a control system for controlling a positioning system. Modern control systems that control positioning systems make it possible to guide a body, with respect to both position and orientation, along defined trajectories in a reference system. Such positioning systems include industrial robots and machine tools, for example, which usually have several movable components.
A trajectory in space that is to be traversed is interpolated by a multitude of individual motion sets or trajectory segments. As a rule, the orientation trajectories specified cannot be continuously differentiated in the programmed intermediate points. This means that the orientation velocity of the body to be positioned is usually different at the end of the first motion set from the orientation velocity at the start of the immediately following second motion set unless a stop is made at a programmed intermediate point. The requirement imposed on the control system for control of the positioning system is to concatenate successive motion sets in such a way that a smooth transition from one motion set to the next occurs with respect to the velocity as well as position. Within a region determined by means of a concatenation criterion, the control system should replace the two motion sets with a concatenation contour so that the transitional motion is continuous in both orientation and velocity of orientation. For the mathematical description of the orientation, use is often made of roll, pitch and yaw angles. (See Paul, R. B., "Robot Manipulators: Mathematics, Programming and Control", and particularly chapter 2, p. 45 et seq. [The MIT Press, Cambridge, Mass., and London, England, 1981].)
In his article, "Planning and Execution of Straight-Line Manipulator Trajectories", published in "Robot Motion" (The MIT Press, 1982), Russell H. Taylor describes an orientation interpolation with continuous transition in which the orientation guidance of the tool in the individual sets is performed through a rotating vector for all tool axes and the concatenating motion is generated through superposition of successive individual sets. In the orientation guidance in the individual sets, advantage is taken of the fact that every three-dimensional rotation can be represented geometrically by a specific rotating vector and an angle of rotation.
Although continuous orientation guidance is obtained by the method described, individual-set interpolation has considerable shortcomings. Depending on the variation of the roll-pitch-yaw angle (hereinafter called RPY angle) C, the tool axes will cut out large or small circles on the unit sphere about the tool tip. In extreme cases, this may give rise to a kind of wobbling motion that is very undesirable, and since there is only one rotating vector for all tool axes, it is pointless to apply this method to kinematics with fewer than six degrees of freedom.
The present invention is directed to the problem of developing a continuous motion-optimal method for the transition between two motion sets or trajectory segments. The present invention is also directed to the problem of developing such a method which is also meaningfully applicable to kinematics with fewer than six degrees of freedom and which lends itself to being combined with an orientation guidance in the individual sets that is understandable by the user. An example of such an orientation guidance is the orientation guidance through a slewing motion of the longitudinal axis of the tool (the X.sub.WZ axis) and a superimposed rotation described by R. P. Paul in "Robot Manipulators: Mathematics, Programming and Control".